-Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class. These notes expand on *a lot*, and some of this material will be reviewed next week.

+Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class. This present page expands on *a lot*, and some of this material will be reviewed next week.

-Applications

-============

+[Linguistic and Philosophical Applications of the Tools We'll be Studying](/applications)

-We mentioned a number of linguistic and philosophical applications of the tools that we'd be helping you learn in the seminar. (We really do mean "helping you learn," not "teaching you." You'll need to aggressively browse and experiment with the material yourself, or nothing we do in a few two-hour sessions will succeed in inducing mastery of it.)

+[Explanation of the "Damn" example shown in class](/damn)

-From linguistics

-----------------

+Basics of Lambda Calculus

+=========================

+

+The lambda calculus we'll be focusing on for the first part of the course has no types. (Some prefer to say it instead has a single type---but if you say that, you have to say that functions from this type to this type also belong to this type. Which is weird... In fact, though, such types are studied, under the name "recursive type." More about these later in the seminar.)

+Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:

+

+<blockquote>

+<strong>Abstract</strong>: <code>(&lambda;a M)</code>

+</blockquote>

+

+We'll tend to write <code>(&lambda;a M)</code> as just `(\a M)`, so we don't have to write out the markup code for the <code>&lambda;</code>. You can yourself write <code>(&lambda;a M)</code> or `(\a M)` or `(lambda a M)`.

+

+<blockquote>

+<strong>Application</strong>: <code>(M N)</code>

+</blockquote>

+

+

+Examples of expressions:

+

+ x

+ (y x)

+ (x x)

+ (\x y)

+ (\x x)

+ (\x (\y x))

+ (x (\x x))

+ ((\x (x x)) (\x (x x)))

+

+The lambda calculus has an associated proof theory. For now, we can regard the

+proof theory as having just one rule, called the rule of **beta-reduction** or

+"beta-contraction". Suppose you have some expression of the form:

+

+ ((\a M) N)

+

+that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.

+

+The rule of beta-reduction permits a transition from that expression to the following:

+

+ M [a:=N]

+

+What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.

+

+What is a free occurrence?

+

+> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.

+

+> If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`.

+

+> An occurrence of a variable is **free** if it's not bound.

+

+For instance:

+

+

+> T is defined to be `(x (\x (\y (x (y z)))))`

+

+The first occurrence of `x` in T is free. The `\x` we won't regard as containing an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free.

-* generalized quantifiers are a special case of operating on continuations

+To read further:

-* (Chris: fill in other applications...)

+* [[!wikipedia Free variables and bound variables]]

-* expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the [summary](/damn) for more explanation and elaboration

+Here's an example of beta-reduction:

+

+ ((\x (y x)) z)

+

+beta-reduces to:

+

+ (y z)

+

+We'll write that like this:

+

+ ((\x (y x)) z) ~~> (y z)

+

+Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write:

+

+ M ~~> N

+

+We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol <code><big><big>&rarr;</big></big></code> for one-step contraction, and the symbol <code><big><big>&#8608;</big></big></code> for zero-or-more step reduction. Hindley and Seldin use <code><big><big><big>&#8883;</big></big></big><sub>1</sub></code> and <code><big><big><big>&#8883;</big></big></big></code>.

+

+When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **beta-convertible**. We'll write that like this:

+

+ M <~~> N

+

+This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.)

+

+In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol <code>&equiv;</code> for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing the metatheory. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in:

+

+> T is defined to be `(M N)`.

+

+We'll regard the following two expressions:

+

+ (\x (x y))

-From philosophy

----------------

+ (\z (z y))

-* the natural semantics for positive free logic is thought by some to have objectionable ontological commitments; Jim says that thought turns on not understanding the notion of a "union type", and conflating the folk notion of "naming" with the technical notion of semantic value. We'll discuss this in due course.

+as syntactically equivalent, since they only involve a typographic change of a bound variable. Read Hankin section 2.3 for discussion of different attitudes one can take about this.

-* those issues may bear on Russell's Gray's Elegy argument in "On Denoting"

+Note that neither of those expressions are identical to:

-* and on discussion of the difference between the meaning of "is beautiful" and "beauty," and the difference between the meaning of "that snow is white" and "the proposition that snow is white."

+ (\x (x w))

-* the apparatus of monads, and techniques for statically representing the semantics of an imperatival language quite generally, are explicitly or implicitly invoked in dynamic semantics

-* the semantics for mutation will enable us to make sense of a difference between numerical and qualitative identity---for purely mathematical objects!

+ (\y (y y))

-* issues in that same neighborhood will help us better understand proposals like Kit Fine's that semantics is essentially coordinated, and that `R a a` and `R a b` can differ in interpretation even when `a` and `b` don't

+because here the second occurrence of `y` is no longer free.

+There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we've linked to about the lambda calculus. We expect you have a good intuitive understanding of what to do already, though, even if you're not able to articulate it rigorously.

+* [More discussion in week 2 notes](/week2/#index1h1)

+

+

+Shorthand

+---------

+

+The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)

+

+

+**Parentheses** Outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:

+

+ (\x (x y))

+

+as:

+

+ \x (x y)

+

+but you should include the parentheses in:

+

+ (\x (x y)) z

+

+and:

+

+ z (\x (x y))

+

+

+**Dot notation** Dot means "put a left paren here, and put the right

+paren as far the right as possible without creating unbalanced

+parentheses". So:

+

+ \x (\y (x y))

+

+can be abbreviated as:

+

+ \x (\y. x y)

+

+and that as:

+

+ \x. \y. x y

+

+This:

+

+ \x. \y. (x y) x

+

+abbreviates:

+

+ \x (\y ((x y) x))

+

+This on the other hand:

+

+ (\x. \y. (x y)) x

+

+abbreviates:

+

+ ((\x (\y (x y))) x)

+

+

+**Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:

+

+ (\x y. M)

+

+Similarly, `(\x (\y (\z M)))` can be abbreviated as:

+

+ (\x y z. M)

+

+

+Lambda terms represent functions

+--------------------------------

+

+The untyped lambda calculus is Turing complete: all (recursively computable) functions can be represented by lambda terms. For some lambda terms, it is easy to see what function they represent:

+

+> `(\x x)` represents the identity function: given any argument `M`, this function

+simply returns `M`: `((\x x) M) ~~> M`.

+

+> `(\x (x x))` duplicates its argument:

+`((\x (x x)) M) ~~> (M M)`

+

+> `(\x (\y x))` throws away its second argument:

+`(((\x (\y x)) M) N) ~~> M`

+

+and so on.

+

+It is easy to see that distinct lambda expressions can represent the same

+function, considered as a mapping from input to outputs. Obviously:

+

+ (\x x)

+

+and:

+

+ (\z z)

+

+both represent the same function, the identity function. However, we said above that we would be regarding these expressions as synactically equivalent, so they aren't yet really examples of *distinct* lambda expressions representing a single function. However, all three of these are distinct lambda expressions:

+

+ (\y x. y x) (\z z)

+

+ (\x. (\z z) x)

+

+ (\z z)

+

+yet when applied to any argument M, all of these will always return M. So they have the same extension. It's also true, though you may not yet be in a position to see, that no other function can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.

+

+The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as proof-theoretically equivalent even though they're not syntactically identical. However, the proof theory we've given so far doesn't permit you to reduce the second expression to the third. So these lambda expressions are non-equivalent.

+

+There's an extension of the proof-theory we've presented so far which does permit this further move. And in that extended proof theory, all computable functions with the same extension do turn out to be equivalent (convertible). However, at that point, we still won't be working with the traditional mathematical notion of a function as a set of ordered pairs. One reason is that the latter but not the former permits many uncomputable functions. A second reason is that the latter but not the former prohibits functions from applying to themselves. We discussed this some at the end of Monday's meeting (and further discussion is best pursued in person).

+

+

+

+Booleans and pairs

+==================

+

+Our definition of these is reviewed in [[Assignment1]].

+

+

+It's possible to do the assignment without using a Scheme interpreter, however

+you should take this opportunity to [get Scheme installed on your

+computer](/how_to_get_the_programming_languages_running_on_your_computer), and

+[get started learning Scheme](/learning_scheme). It will help you test out

+proposed answers to the assignment.

+

+

+There's also a (slow, bare-bones, but perfectly adequate) version of Scheme available for online use at <http://tryscheme.sourceforge.net/>.

@@ -57,6+269,14 @@ It's possible to enhance the lambda calculus so that functions do get identified

It's often said that dynamic systems are distinguished because they are the ones in which **order matters**. However, there are many ways in which order can matter. If we have a trivalent boolean system, for example---easily had in a purely functional calculus---we might choose to give a truth-table like this for "and":

true and true = true

+ true and * = *

+ true and false = false

+ * and true = *

+ * and * = *

+ * and false = *

+ false and true = false

+ false and * = false

+ false and false = false

And then we'd notice that `* and false` has a different intepretation than `false and *`. (The same phenomenon is already present with the material conditional in bivalent logics; but seeing that a non-symmetric semantics for `and` is available even for functional languages is instructive.)

@@ -85,7+305,7 @@ Finally, you'll see the term **dynamic** used in a variety of ways in the litera

@@ -93,6+313,16 @@ Finally, you'll see the term **dynamic** used in a variety of ways in the litera

For the most part, these uses are only loosely connected to each other. We'll tend to use "imperatival" to describe the kinds of semantic properties made available in dynamic semantics, languages which have robust notions of sequencing changes, and so on.

+To read further about the relation between declarative or functional programming, on the one hand, and imperatival programming on the other, you can begin here:

Here's how it looks to say the same thing in various of these languages.

-1. Binding suitable values to the variables `three` and `two`, and adding them.

+The following site may be useful; it lets you run a Scheme interpreter inside your web browser: [Try Scheme in your web browser](http://tryscheme.sourceforge.net/). See also our links about [[learning Scheme]] and [[learning OCaml]].

+

+&nbsp;

+

+1. Function application and parentheses

+

+ In Scheme and the lambda calculus, the functions you're applying always go to the left. So you write `(foo 2)` and also `(+ 2 3)`.

+

+ Mostly that's how OCaml is written too:

+

+ foo 2

+

+ But a few familiar binary operators can be written infix, so:

+

+ 2 + 3

+

+ You can also write them operator-leftmost, if you put them inside parentheses to help the parser understand you:

+

+ ( + ) 2 3

+

+ I'll mostly do this, for uniformity with Scheme and the lambda calculus.

+

+ In OCaml and the lambda calculus, this:

+

+ foo 2 3

+

+ means the same as:

+

+ ((foo 2) 3)

+

+ These functions are "curried". `foo 2` returns a `2`-fooer, which waits for an argument like `3` and then foos `2` to it. `( + ) 2` returns a `2`-adder, which waits for an argument like `3` and then adds `2` to it. For further reading:

+

+* [[!wikipedia Currying]]

+

+ In Scheme, on the other hand, there's a difference between `((foo 2) 3)` and `(foo 2 3)`. Scheme distinguishes between unary functions that return unary functions and binary functions. For our seminar purposes, it will be easiest if you confine yourself to unary functions in Scheme as much as possible.

+

+ Scheme is very sensitive to parentheses and whenever you want a function applied to any number of arguments, you need to wrap the function and its arguments in a parentheses. So you have to write `(foo 2)`; if you only say `foo 2`, Scheme won't understand you.

+

+ Scheme uses a lot of parentheses, and they are always significant, never optional. Often the parentheses mean "apply this function to these arguments," as just described. But in a moment we'll see other constructions in Scheme where the parentheses have different roles. They do lots of different work in Scheme.

+

+

+2. Binding suitable values to the variables `three` and `two`, and adding them.

In Scheme:

(let* ((three 3))

- (let ((two 2))

+ (let* ((two 2))

(+ three two)))

- In OCaml:

+ Most of the parentheses in this construction *aren't* playing the role of applying a function to some arguments---only the ones in `(+ three two)` are doing that.

- let three = 3 in

- let two = 2 in

- three + two

- Notice OCaml lets you write the `+` in between the `three` and `two`, as you're accustomed to. However most functions need to come leftmost, even if they're binary. And you can do this with `+` too, if you enclose it in parentheses so that the OCaml parser doesn't get confused by your syntax:

+ In OCaml:

let three = 3 in

let two = 2 in

( + ) three two

- In the lambda calculus: here we're on our own, we don't have predefined constants like `+` and `3` and `2` to work with. We've got to build up everything from scratch. We'll be seeing how to do that over the next weeks.

+ In the lambda calculus:

+

+ Here we're on our own, we don't have predefined constants like `+` and `3` and `2` to work with. We've got to build up everything from scratch. We'll be seeing how to do that over the next weeks.

But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in:

@@ -179,23+450,27 @@ Here's how it looks to say the same thing in various of these languages.

Of course you don't need to remember any of this syntax. We're just illustrating it so that you see that in Scheme and OCaml it looks somewhat different than we had above. The difference is much more obvious than it is in C.

- In the lambda calculus: sorry, you can't do mutation. At least, not natively. Later in the term we'll be learning how in fact, really, you can embed mutation inside the lambda calculus even though the lambda calculus has no primitive facilities for mutation.

+ In the lambda calculus:

+ Sorry, you can't do mutation. At least, not natively. Later in the term we'll be learning how in fact, really, you can embed mutation inside the lambda calculus even though the lambda calculus has no primitive facilities for mutation.

3. Anonymous functions

- Functions are "first-class values" in the lambda calculus, in Scheme, and in OCaml. What that means is that they can be arguments to other functions. They can be the results of the application of other functions to some arguments. They can be stored in data structures. And so on.

+ Functions are "first-class values" in the lambda calculus, in Scheme, and in OCaml. What that means is that they can be arguments to, and results of, other functions. They can be stored in data structures. And so on. To read further:

+

+ * [[!wikipedia Higher-order function]]

+ * [[!wikipedia First-class function]]

- First, we'll show what "anonymous" functions look like. These are functions that have not been bound as values to any variables. That is, there are no variables whose value they are.

+ We'll begin by looking at what "anonymous" functions look like. These are functions that have not been bound as values to any variables. That is, there are no variables whose value they are.

In the lambda calculus:

(\x M)

- is always anonymous! Here `M` stands for any expression of the language, simple or complex. It's only when you do

+ ---where `M` is any simple or complex expression---is anonymous. It's only when you do:

((\y N) (\x M))

@@ -205,29+480,14 @@ Here's how it looks to say the same thing in various of these languages.

(lambda (x) M)

- Not very different, right? For example, if `M` stands for `(+ 3 x)`, then this is an anonymous function that adds 3 to whatever argument it's given:

+ Not very different, right? For example, if `M` stands for `(+ 3 x)`, then here is an anonymous function that adds 3 to whatever argument it's given:

(lambda (x) (+ 3 x))

- Scheme uses a lot of parentheses, and they are always significant, never optional. In `(+ 3 x)` the parentheses mean "apply the function `+` to the arguments `3` and `x`. In `(lambda (x) ...)` the parentheses have a different meaning: they mark where the anonymous function you're defining begins and ends, and so on. As you'll see, parentheses have yet further roles in Scheme. I know it's confusing.

-

In OCaml, we write our anonymous function like this:

- fun x -> (3 + x)

-

- or:

-

- fun x -> (( + ) 3 x)

-

- In OCaml, parentheses only serve a grouping function and they often can be omitted. Or more could be added. For instance, we could equally well say:

-

fun x -> ( + ) 3 x

- or:

-

- (fun x -> (( + ) (3) (x)))

-

- As we saw above, parentheses can often be omitted in the lambda calculus too. But not in Scheme. Every parentheses has a specific role.

4. Supplying an argument to an anonymous function

@@ -235,7+495,7 @@ Here's how it looks to say the same thing in various of these languages.

@@ -247,7+507,7 @@ Here's how it looks to say the same thing in various of these languages.

Let's go back and re-consider this Scheme expression:

(let* ((three 3))

- (let ((two 2))

+ (let* ((two 2))

(+ three two)))

Scheme also has a simple `let` (without the ` *`), and it permits you to group several variable bindings together in a single `let`- or `let*`-statement, like this:

@@ -258,23+518,23 @@ Here's how it looks to say the same thing in various of these languages.

Often you'll get the same results whether you use `let*` or `let`. However, there are cases where it makes a difference, and in those cases, `let*` behaves more like you'd expect. So you should just get into the habit of consistently using that. It's also good discipline for this seminar, especially while you're learning, to write things out the longer way, like this:

(let* ((three 3))

- (let ((two 2))

+ (let* ((two 2))

(+ three two)))

However, here you've got the double parentheses in `(let* ((three 3)) ...)`. They're doubled because the syntax permits more assignments than just the assignment of the value `3` to the variable `three`. Myself I tend to use `[` and `]` for the outer of these parentheses: `(let* [(three 3)] ...)`. Scheme can be configured to parse `[...]` as if they're just more `(...)`.

- Someone asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:

+ It was asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:

(let* [(three (+ 1 2))]

- (let [(two 2)]

+ (let* [(two 2)]

(+ three two)))

- The question also came up whether the `(+ 1 2)` computation would be performed before or after it was bound to the variable `three`. That's a terrific question. Let's say this: both strategies could be reasonable designs for a language. We are going to discuss this carefully in coming weeks. In fact Scheme and OCaml make the same design choice. But you should think of the underlying form of the `let`-statement as not settling this by itself.

+ It was also asked whether the `(+ 1 2)` computation would be performed before or after it was bound to the variable `three`. That's a terrific question. Let's say this: both strategies could be reasonable designs for a language. We are going to discuss this carefully in coming weeks. In fact Scheme and OCaml make the same design choice. But you should think of the underlying form of the `let`-statement as not settling this by itself.

Repeating our starting point for reference:

(let* [(three 3)]

- (let [(two 2)]

+ (let* [(two 2)]

(+ three two)))

Recall in OCaml this same computation was written:

@@ -301,7+561,7 @@ Here's how it looks to say the same thing in various of these languages.

(let* [(bar (lambda (x) B))] M)

- then wherever `bar` occurs in `M` (and isn't rebound by a more local "let" or "lambda"), it will be interpreted as the function `(lambda (x) B)`.

+ then wherever `bar` occurs in `M` (and isn't rebound by a more local `let` or `lambda`), it will be interpreted as the function `(lambda (x) B)`.

Similarly, in OCaml:

@@ -316,7+576,7 @@ Here's how it looks to say the same thing in various of these languages.

((lambda (bar) (bar A)) (lambda (x) B))

- which, as we'll see, is equivalent to:

+ which beta-reduces to:

((lambda (x) B) A)

@@ -361,8+621,7 @@ Here's how it looks to say the same thing in various of these languages.

let x = A;;

... rest of the file or interactive session ...

- It's easy to be lulled into thinking this is a kind of imperative construction. *But it's not!* It's really just a shorthand for the compound "let"-expressions we've already been looking at, taking the maximum syntactically permissible scope. (Compare the "dot" convention in the lambda calculus, discussed above.)

-

+ It's easy to be lulled into thinking this is a kind of imperative construction. *But it's not!* It's really just a shorthand for the compound `let`-expressions we've already been looking at, taking the maximum syntactically permissible scope. (Compare the "dot" convention in the lambda calculus, discussed above. I'm fudging a bit here, since in Scheme `(define ...)` is really shorthand for a `letrec` epression, which we'll come to in later classes.)

9. Some shorthand

@@ -434,17+693,20 @@ Here's how it looks to say the same thing in various of these languages.

and there's no more mutation going on there than there is in:

- <pre>

- <code>&forall;x. (F x or &forall;x (not (F x)))</code>

- </pre>

+ <pre><code>&forall;x. (F x or &forall;x (not (F x)))

+ </code></pre>

When a previously-bound variable is rebound in the way we see here, that's called **shadowing**: the outer binding is shadowed during the scope of the inner binding.

- These are the same as: `((foo 2) 3)`. In other words, functions in OCaml are "curried". `foo 2` returns a `2`-fooer, which waits for an argument like `3` and then foos `2` to it. `( + ) 2` returns a `2`-adder, which waits for an argument like `3` and then adds `2` to it.

-

- In Scheme, on the other hand, there's a difference between `((foo 2) 3)` and `(foo 2 3)`. Scheme distinguishes between unary functions that return unary functions and binary functions. For our seminar purposes, it will be easiest if you confine yourself to unary functions in Scheme as much as possible.

-

- Additionally, as said above, Scheme is very sensitive to parentheses and whenever you want a function applied to any number of arguments, you need to wrap the function and its arguments in a parentheses.

-don't involve any changes or sequencing in the sense we're trying to identify. As we said, these programs are just syntactic variants of (single) compound syntactic structures involving "let"s and "lambda"s.

+don't involve any changes or sequencing in the sense we're trying to identify. As we said, these programs are just syntactic variants of (single) compound syntactic structures involving `let`s and `lambda`s.

Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:

-The lambda calculus we'll be focusing on for the first part of the course has no types. (Some prefer to say it instead has a single type---but if you say that, you have to say that functions from this type to this type also belong to this type. Which is weird.)

-Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:

-

-<blockquote>

-<strong>Abstract</strong>: <code>(&lambda;a M)</code>

-</blockquote>

-

-We'll tend to write <code>(&lambda;a M)</code> as just `(\a M)`, so we don't have to write out the markup code for the <code>&lambda;</code>. You can yourself write <code>(&lambda;a M)</code> or `(\a M)` or `(lambda a M)`.

-

-<blockquote>

-<strong>Application</strong>: <code>(M N)</code>

-</blockquote>

-

-Some authors reserve the term "term" for just variables and abstracts. We won't participate in that convention; we'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.

-

-Examples of expressions:

-

- x

- (y x)

- (x x)

- (\x y)

- (\x x)

- (\x (\y x))

- (x (\x x))

- ((\x (x x)) (\x (x x)))

-

-The lambda calculus has an associated proof theory. For now, we can regard the proof theory as having just one rule, called the rule of **beta-reduction** or "beta-contraction". Suppose you have some expression of the form:

-

- ((\a M) N)

-

-that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.

-

-The rule of beta-reduction permits a transition from that expression to the following:

-

- M [a:=N]

-

-What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.

-

-What is a free occurrence?

-

-> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.

-

-> If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`.

-

-> An occurrence of a variable is **free** if it's not bound.

-

-For instance:

-

-

-> T is defined to be `(x (\x (\y (x (y z)))))`

-

-The first occurrence of `x` in T is free. The `\x` we won't regard as being an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free.

-

-Here's an example of beta-reduction:

-

- ((\x (y x)) z)

-

-beta-reduces to:

-

- (y z)

-

-We'll write that like this:

-

- ((\x (y x)) z) ~~> (y z)

-

-Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write:

-

- M ~~> N

-

-We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol <code><big><big>&rarr;</big></big></code> for one-step contraction, and the symbol <code><big><big>&#8608;</big></big></code> for zero-or-more step reduction. Hindley and Seldin use <code><big><big><big>&#8883;</big></big></big><sub>1</sub></code> and <code><big><big><big>&#8883;</big></big></big></code>.

-

-When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **beta-convertible**. We'll write that like this:

-

- M <~~> N

-

-This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.)

-

-In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol <code>&equiv;</code> for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing metatheory for the lambda calculus. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in:

-

-> T is defined to be `(M N)`.

-

-We'll regard the following two expressions:

-

- (\x (x y))

-

- (\z (z y))

-

-as syntactically equivalent, since they only involve a typographic change of a bound variable. Read Hankin section 2.3 for discussion of different attitudes one can take about this.

-There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we've linked to about the lambda calculus. We expect you have a good intuitive understanding of what to do already, though, even if you're not able to articulate it rigorously.

-

-

-Shorthand

----------

-

-The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)

-

-

-**Dot notation** Dot means "put a left paren here, and put the right

-paren as far the right as possible without creating unbalanced

-parentheses". So:

-

- (\x (\y (x y)))

-

-can be abbreviated as:

-

- (\x (\y. x y))

-

-and:

-

- (\x (\y. (z y) z))

-

-would abbreviate:

-

- (\x (\y ((z y) z)))

-

-This on the other hand:

-

- (\x (\y. z y) z)

-

-would abbreviate:

-

- (\x (\y (z y)) z)

-

-**Parentheses** Outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:

-

- (\x. x y)

-

-as:

-

- \x. x y

-

-but you should include the parentheses in:

-

- (\x. x y) z

-

-and:

-

- z (\x. x y)

-

-**Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:

-

- (\x y. M)

-

-Similarly, `(\x (\y (\z M)))` can be abbreviated as:

-

- (\x y z. M)

-

-

-Lambda terms represent functions

---------------------------------

-

-All (recursively computable) functions can be represented by lambda

-terms (the untyped lambda calculus is Turing complete). For some lambda terms, it is easy to see what function they represent:

-

-> `(\x x)` represents the identity function: given any argument `M`, this function

-simply returns `M`: `((\x x) M) ~~> M`.

-

-> `(\x (x x))` duplicates its argument:

-`((\x (x x)) M) ~~> (M M)`

-

-> `(\x (\y x))` throws away its second argument:

-`(((\x (\y x)) M) N) ~~> M`

-

-and so on.

-

-It is easy to see that distinct lambda expressions can represent the same

-function, considered as a mapping from input to outputs. Obviously:

-

- (\x x)

-

-and:

-

- (\z z)

-

-both represent the same function, the identity function. However, we said above that we would be regarding these expressions as synactically equivalent, so they aren't yet really examples of *distinct* lambda expressions representing a single function. However, all three of these are distinct lambda expressions:

-

- (\y x. y x) (\z z)

-

- (\x. (\z z) x)

-

- (\z z)

-

-yet when applied to any argument M, all of these will always return M. So they have the same extension. It's also true, though you may not yet be in a position to see, that no other function can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.

-

-The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as proof-theoretically equivalent even though they're not syntactically identical. However, the proof theory we've given so far doesn't permit you to reduce the second expression to the third. So these lambda expressions are non-equivalent.

-

-There's an extension of the proof-theory we've presented so far which does permit this further move. And in that extended proof theory, all computable functions with the same extension do turn out to be equivalent (convertible). However, at that point, we still won't be working with the traditional mathematical notion of a function as a set of ordered pairs. One reason is that the latter but not the former permits uncomputable functions. A second reason is that the latter but not the former prohibits functions from applying to themselves. We discussed this some at the end of Monday's meeting (and further discussion is best pursued in person).

-

-

-

-Booleans and pairs

-==================

-

-Our definition of these is reviewed in [[Assignment1]].

-

-

-It's possible to do the assignment without using a Scheme interpreter, however

-you should take this opportunity to [get Scheme installed on your

-computer](/how_to_get_the_programming_languages_running_on_your_computer), and

-[get started learning Scheme](/learning_scheme). It will help you test out